Find the volume of the solid cut from the first octant by the surface z = 4 - x² - y.

Steps to Solve

1. Identify the limits of integration

The equation for the surface is given by $z = 4 - x^2 - y$. To find the volume in the first octant, set $z = 0$ to find the boundary in the $xy$-plane: $$ 0 = 4 - x^2 - y \implies y = 4 - x^2 $$ Next, determine where $y$ is non-negative: $4 - x^2 \geq 0$, giving $x^2 \leq 4$, or $-2 \leq x \leq 2$. Since we are in the first octant, $0 \leq x \leq 2$.

2. Set up the double integral for volume

The volume under the surface above the $xy$-plane can be represented as a double integral: $$ V = \int_0^2 \int_0^{4 - x^2} z , dy , dx $$ Here, the outer integral integrates with respect to $x$ and the inner integral integrates with respect to $y$.

3. Evaluate the inner integral

Now perform the inner integral: $$ \int_0^{4 - x^2} (4 - x^2 - y) , dy $$ Calculate this: $$ = [ (4 - x^2)y - \frac{y^2}{2} ]_0^{4 - x^2} $$ Substituting the limits: $$ = (4 - x^2)(4 - x^2) - \frac{(4 - x^2)^2}{2} $$

4. Evaluate the outer integral

Next, evaluate the outer integral: $$ V = \int_0^2 \left( (4 - x^2)(4 - x^2) - \frac{(4 - x^2)^2}{2} \right) , dx $$ Simplify the expression and compute this integral.

5. Final value of the volume

After evaluating the outer integral, you will obtain the volume of the solid cut from the first octant.